arXiv subjects

George Raptis

Publications and source records attributed to George Raptis.

26 records · Page 2Linked to original sources

arXiv2017

Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem

We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we prove that such a bivariant transformation is uniquely determined by its value at the universal disk bundle. This description of bivariant transformations yields a short proof of the Dwyer-Weiss-Williams family index theorem for the parametrized A-theory Euler characteristic of a smooth bundle.

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arXiv2017

A remark on configuration spaces of two points

We prove a homotopy invariance result for a certain covering space of the space of ordered configurations of two points in $M \times X$ where $M$ is a closed smooth manifold and $X$ is any fixed aspherical space which is not a point.

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arXiv2016

K-theory of derivators revisited

We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.

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arXiv2012

On the map of Bökstedt-Madsen from the cobordism category to $A$-theory

Bökstedt and Madsen defined an infinite loop map from the embedded $d$-dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic $K$-theory of $BO(d)$ in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it extends the universal parametrized $A$-theory Euler characteristic of smooth bundles with compact $d$-dimensional fibers, as defined by Dwyer, Weiss and Williams. The second result is that it actually factors through the canonical unit map $Q(BO(d)_+) \to A(BO(d))$.

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arXiv2011

A note on K-theory and triangulated derivators

In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis's comparison and localization conjectures for derivator K-theory cannot be simultaneously true.

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arXiv2011

On the homotopy type of certain cobordism categories of surfaces

Let $\mathcal{A}_{g,d}$ be the (topological) cobordism category of orientable surfaces whose connected components are homeomorphic to either $S^1 \times I$ with one incoming and one outgoing boundary component or the surface $Σ_{g,d}$ of genus $g$ and $d$ boundary components that are all incoming. In this paper, we study the homotopy type of the classifying space of the cobordism categories $\mathcal{A}_{g,d}$ and the associated (ordinary) cobordism categories of their connected components $\mathbb{A}_d$. $\mathcal{A}_{0,2}$ is the cobordism category of complex annuli that was considered by Costello and $\mathbb{A}_2$ is homotopy equivalent with the positive boundary 1-dimensional embedded cobordism category of Galatius-Madsen-Tillmann-Weiss. We identify their homotopy type with the infinite loop spaces associated with certain Thom spectra.

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arXiv2011

Simplicial presheaves of coalgebras

The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the local weak equivalences of the underlying simplicial presheaves. This model category is naturally linked to the R-local homotopy theory of simplicial presheaves and the homotopy theory of simplicial R-modules by Quillen adjunctions. We study the comparison with the R-local homotopy category of simplicial presheaves in the special case where R is a presheaf of algebraically closed (or perfect) fields. If R is a presheaf of algebraically closed fields, we show that the R-local homotopy category of simplicial presheaves embeds fully faithfully in the homotopy category of simplicial R-coalgebras.

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arXiv2009

On the Cofibrant Generation of Model Categories

The paper studies the problem of the cofibrant generation of a model category. We prove that, assuming Vopěnka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial model category. We discuss cases where this result implies that the class of weak equivalences in a cofibrantly generated model category is accessibly embedded. We also prove a necessary condition for a model category to be cofibrantly generated by a set of generating cofibrations between cofibrant objects.

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